First order logic
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First-Order Logic. Chapter 8. Outline. Why FOL? Syntax and semantics of FOL Using FOL Wumpus world in FOL Knowledge engineering in FOL. Pros and cons of propositional logic. Propositional logic is declarative Propositional logic allows partial/disjunctive/negated information

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First-Order Logic

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First order logic

First-Order Logic

Chapter 8


Outline

Outline

  • Why FOL?

  • Syntax and semantics of FOL

  • Using FOL

  • Wumpus world in FOL

  • Knowledge engineering in FOL


Pros and cons of propositional logic

Pros and cons of propositional logic

  • Propositional logic is declarativePropositional logic allows partial/disjunctive/negated information

    • (unlike most data structures and databases)

  • Propositional logic is compositional:

    • meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2

  • Meaning in propositional logic is context-independent

    • (unlike natural language, where meaning depends on context)

  • Propositional logic has very limited expressive power

    • (unlike natural language)

    • E.g., cannot say "pits cause breezes in adjacent squares“

      • except by writing one sentence for each square


The others

The Others


First order logic1

First-order logic

  • Whereas propositional logic assumes the world contains facts,

  • first-order logic (like natural language) assumes the world contains

    • Objects: people, houses, numbers, colors, baseball games, wars, …Relations: red, round, prime, brother of, bigger than, part of, comes between, …

    • Functions: father of, best friend, one more than, plus, …


Syntax of fol basic elements

Syntax of FOL: Basic elements

  • ConstantsKingJohn, 2, NUS,...

  • PredicatesBrother, >,...

  • FunctionsSqrt, LeftLegOf,...

  • Variablesx, y, a, b,...

  • Connectives, , , , 

  • Equality=

  • Quantifiers , 


Atomic sentences

Atomic sentences

Atomic sentence =predicate (term1,...,termn) or term1 = term2

Term =function (term1,...,termn) or constant or variable

  • E.g., Brother(KingJohn,RichardTheLionheart) > (Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn)))


Complex sentences

Complex sentences

  • Complex sentences are made from atomic sentences using connectives

    S, S1 S2, S1  S2, S1 S2, S1S2,

    E.g. Sibling(KingJohn,Richard)  Sibling(Richard,KingJohn)

    >(1,2)  ≤ (1,2)

    >(1,2)  >(1,2)


Syntax of fol

Syntax of FOL

Syntax of Propositional Logic


Truth in first order logic

Truth in first-order logic

  • Sentences are true with respect to a model and an interpretation

  • Model contains objects (domainelements) and relations among them

  • Interpretation specifies referents for

    constantsymbols→objects

    predicatesymbols → relations

    functionsymbols →functional relations

  • An atomic sentence predicate(term1,...,termn) is true

    iff the objects referred to by term1,...,termn

    are in the relation referred to by predicate


Models for fol example

Models for FOL: Example


Universal quantification

Universal quantification

  • <variables> <sentence>

    Everyone at NUS is smart:

    x At(x,NUS)  Smart(x)

  • x P is true in a model m iff P is true with x being each possible object in the model

  • Roughly speaking, equivalent to the conjunction of instantiations of P

    At(KingJohn,NUS)  Smart(KingJohn)

    At(Richard,NUS)  Smart(Richard)

    At(NUS,NUS)  Smart(NUS)

     ...


A common mistake to avoid

A common mistake to avoid

  • Typically,  is the main connective with 

  • Common mistake: using  as the main connective with :

    x At(x,NUS)  Smart(x)

    means “Everyone is at NUS and everyone is smart”


Existential quantification

Existential quantification

  • <variables> <sentence>

  • Someone at NUS is smart:

  • x At(x,NUS)  Smart(x)$

  • xP is true in a model m iff P is true with x being some possible object in the model

  • Roughly speaking, equivalent to the disjunction of instantiations of P

    At(KingJohn,NUS)  Smart(KingJohn)

    At(Richard,NUS)  Smart(Richard)

    At(NUS,NUS)  Smart(NUS)

     ...


Another common mistake to avoid

Another common mistake to avoid

  • Typically,  is the main connective with 

  • Common mistake: using  as the main connective with :

    x At(x,NUS)  Smart(x)

    is true if there is anyone who is not at NUS!


Properties of quantifiers

Properties of quantifiers

  • x y is the same as yx

  • x y is the same as yx

  • x y is not the same as yx

  • x y Loves(x,y)

    • “There is a person who loves everyone in the world”

  • yx Loves(x,y)

    • “Everyone in the world is loved by at least one person”

  • Quantifier duality: each can be expressed using the other

  • x Likes(x,IceCream)x Likes(x,IceCream)

  • x Likes(x,Broccoli) xLikes(x,Broccoli)


Equality

Equality

  • term1 = term2is true under a given interpretation if and only if term1and term2refer to the same object

  • E.g., definition of Sibling in terms of Parent:

    x,ySibling(x,y)  [(x = y)  m,f  (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)]


Using fol

Using FOL

The kinship domain:Brothers are siblings

x,y Brother(x,y) Sibling(x,y)

  • One's mother is one's female parent

    m,c Mother(c) = m (Female(m) Parent(m,c))

  • “Sibling” is symmetric

    x,y Sibling(x,y) Sibling(y,x)


Using fol1

Using FOL

The set domain:

  • s Set(s)  (s = {} )  (x,s2 Set(s2)  s = {x|s2})

  • x,s {x|s} = {}

  • x,s x  s  s = {x|s}

  • x,s x  s  [ y,s2} (s = {y|s2}  (x = y  x  s2))]

  • s1,s2 s1 s2 (x x  s1 x  s2)

  • s1,s2 (s1 = s2)  (s1 s2 s2 s1)

  • x,s1,s2 x  (s1 s2)  (x  s1 x  s2)

  • x,s1,s2 x  (s1 s2)  (x  s1 x  s2)


Interacting with fol kbs

Interacting with FOL KBs

  • Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5:

    Tell(KB,Percept([Smell,Breeze,None],5))

    Ask(KB,a BestAction(a,5))

  • I.e., does the KB entail some best action at t=5?

  • Answer: Yes, {a/Shoot} ← substitution (binding list)

  • Given a sentence S and a substitution σ,

  • Sσ denotes the result of plugging σ into S; e.g.,

    S = Smarter(x,y)

    σ = {x/Hillary,y/Bill}

    Sσ = Smarter(Hillary,Bill)

  • Ask(KB,S) returns some/all σ such that KB╞ σ


Knowledge base for the wumpus world

Knowledge base for the wumpus world

  • Perception

    • t,s,b Percept([s,b,Glitter],t)  Glitter(t)

  • Reflex

    • t Glitter(t)  BestAction(Grab,t)


Deducing hidden properties

Deducing hidden properties

  • x,y,a,b Adjacent([x,y],[a,b]) 

    [a,b]  {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}

    Properties of squares:

  • s,t At(Agent,s,t)  Breeze(t)  Breezy(s)

    Squares are breezy near a pit:

    • Diagnostic rule---infer cause from effect

      s Breezy(s)  \Exi{r} Adjacent(r,s)  Pit(r)$

    • Causal rule---infer effect from cause

      r Pit(r)  [s Adjacent(r,s)  Breezy(s)$ ]


Knowledge engineering in fol

Knowledge engineering in FOL

  • Identify the task

  • Assemble the relevant knowledge

  • Decide on a vocabulary of predicates, functions, and constants

  • Encode general knowledge about the domain

  • Encode a description of the specific problem instance

  • Pose queries to the inference procedure and get answers

  • Debug the knowledge base


The electronic circuits domain

The electronic circuits domain

One-bit full adder


The electronic circuits domain1

The electronic circuits domain

  • Identify the task

    • Does the circuit actually add properly? (circuit verification)

  • Assemble the relevant knowledge

    • Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)

    • Irrelevant: size, shape, color, cost of gates

  • Decide on a vocabulary

    • Alternatives:

      Type(X1) = XOR

      Type(X1, XOR)

      XOR(X1)


The electronic circuits domain2

The electronic circuits domain

  • Encode general knowledge of the domain

    • t1,t2 Connected(t1, t2)  Signal(t1) = Signal(t2)

    • t Signal(t) = 1  Signal(t) = 01 ≠ 0t1,t2 Connected(t1, t2)  Connected(t2, t1)g Type(g) = OR  Signal(Out(1,g)) = 1 n Signal(In(n,g)) = 1g Type(g) = AND  Signal(Out(1,g)) = 0 n Signal(In(n,g)) = 0g Type(g) = XOR  Signal(Out(1,g)) = 1  Signal(In(1,g)) ≠ Signal(In(2,g))g Type(g) = NOT  Signal(Out(1,g)) ≠ Signal(In(1,g))


The electronic circuits domain3

The electronic circuits domain

  • Encode the specific problem instance

    Type(X1) = XOR Type(X2) = XOR

    Type(A1) = AND Type(A2) = AND

    Type(O1) = OR

    Connected(Out(1,X1),In(1,X2))Connected(In(1,C1),In(1,X1))

    Connected(Out(1,X1),In(2,A2))Connected(In(1,C1),In(1,A1))

    Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),In(2,X1))

    Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),In(2,A1))

    Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1),In(2,X2))

    Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1),In(1,A2))


The electronic circuits domain4

The electronic circuits domain

  • Pose queries to the inference procedure

    What are the possible sets of values of all the terminals for the adder circuit?

    i1,i2,i3,o1,o2 Signal(In(1,C_1)) = i1 Signal(In(2,C1)) = i2 Signal(In(3,C1)) = i3 Signal(Out(1,C1)) = o1 Signal(Out(2,C1)) = o2

  • Debug the knowledge base

    May have omitted assertions like 1 ≠ 0


Summary

Summary

  • First-order logic:

    • objects and relations are semantic primitives

    • syntax: constants, functions, predicates, equality, quantifiers

  • Increased expressive power: sufficient to define wumpus world


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